Algebraic Riccati Matrix Equation for Eigen- Decomposition of Special Structured Matrices; Applications in Structural Mechanics
In this paper Algebraic Riccati matrix equation is used for Eigen-decomposition of special structured matrices. This is achieved by similarity transformation and then using algebraic riccati matrix equation to triangulation of matrices. The process is decomposition of matrices into small and specially structured submatrices with low dimensions for fast and easy finding of Eigenpairs. Numerical and structural examples included showing the efficiency of present method.
<p>[1] Bathe KJ, Wilson EL. Numerical Methods for Finite Element Analysis.
Prentice Hall: Englewood Clffis,NJ, 1976.
[2] Livesley RK. Mathematical Methods for Engineers. Ellis Horwood:
Chichester, U.K., 1989.
[3] George J. Simitses, Dewey H. Hodges. Fundamentals of Structural
Stability. Elsevier Inc. 2006.
[4] Jennings A, McKeown JJ. Matrix Computation. Wiley: New York, 1992.
[5] A. Kaveh and H. Rahami, New canonical forms for analytical solution of
problems in structural mechanics, Communications in Numerical
Methods in Engineering, No. 9, 21(2005) 499-513.
[6] Kaveh A., Nouri M. and Taghizadieh N.: Eigensolution for adjacency
and Laplacian matrices of large repetitive structural models. Scientia
Iranica, 16(2009)481-489.
[7] Nouri M.: Free vibration of large regular repetitive structural structures,
International Journal of Science and Engineering Investigations,
Volume 1, Issue 1, 2012, Pages 92-96.
[8] Cuppen, J.J.M. “A divide and conquer method for the symmetric
tridiagonal eigenproblem”, Numerische Mathematik, 36, pp. 177–195
(1981).
[9] Kaveh A., Nouri M. and Taghizadieh N.: An efficient solution method
for the free vibration of large repetitive space structures. Advances in
Structural Engineering, 14(2011)151-161.
[10] Zhou K M, Doyle J, Glover K. Robust and Optimal Control. New
Jersey: Prentice-Hall, 1996.
[11] Lin Z L. Global control of linear systems with saturating actuators.
Automatica, 1998, 34(7): 897-905/
[12] D. A. Bini, B. Iannazzo, B. Meini, and F. Poloni. Nonsymmetric
algebraic Riccati equations associated with an M-matrix: recent
advances and algorithms. In Dagstuhl Seminar Proceedings, "Numerical
Methods for Structured Markov Chains", 07461, 2007.
[13] C.-H. Guo. Nonsymmetric algebraic Riccati equations and Wiener-Hopf
factorization for M-matrices. SIAM J. Matrix Anal. Appl., 23(1):225_242
(electronic), 2001.
[14] Ni M L. A note on the maximum solutions of Riccati equations.
Automatica, 1991, 27(6): 1059-1060.
[15] Ni Mao-Lin. Design of Robust Control Systems: Theory and
Applications (Ph. D. dissertation), Chinese Academy of Space
Technology, 1992 (in Chinese).
[16] Zhou K M, Doyle J, Glover K. Robust and Optimal Control. New
Jersey: Prentice-Hall, 1996.
[17] Kaveh, A. Structural Mechanics: Graph and Matrix Methods, 3rd ed.
Somerset: Research Studies Press, 2004.
[18] A. Kaveh and K. Koohestani, Combinatorial optimization of special
graphs for nodal ordering and graph partitioning, Acta Mechanica, Nos.
(1-2), 207(2009)95-108.
496-501.
[19] Potter, J. E. 1966. Matrix quadratic solutions. SIAM J. Appl. Math. 14:</p>
<p>[1] Bathe KJ, Wilson EL. Numerical Methods for Finite Element Analysis.
Prentice Hall: Englewood Clffis,NJ, 1976.
[2] Livesley RK. Mathematical Methods for Engineers. Ellis Horwood:
Chichester, U.K., 1989.
[3] George J. Simitses, Dewey H. Hodges. Fundamentals of Structural
Stability. Elsevier Inc. 2006.
[4] Jennings A, McKeown JJ. Matrix Computation. Wiley: New York, 1992.
[5] A. Kaveh and H. Rahami, New canonical forms for analytical solution of
problems in structural mechanics, Communications in Numerical
Methods in Engineering, No. 9, 21(2005) 499-513.
[6] Kaveh A., Nouri M. and Taghizadieh N.: Eigensolution for adjacency
and Laplacian matrices of large repetitive structural models. Scientia
Iranica, 16(2009)481-489.
[7] Nouri M.: Free vibration of large regular repetitive structural structures,
International Journal of Science and Engineering Investigations,
Volume 1, Issue 1, 2012, Pages 92-96.
[8] Cuppen, J.J.M. “A divide and conquer method for the symmetric
tridiagonal eigenproblem”, Numerische Mathematik, 36, pp. 177–195
(1981).
[9] Kaveh A., Nouri M. and Taghizadieh N.: An efficient solution method
for the free vibration of large repetitive space structures. Advances in
Structural Engineering, 14(2011)151-161.
[10] Zhou K M, Doyle J, Glover K. Robust and Optimal Control. New
Jersey: Prentice-Hall, 1996.
[11] Lin Z L. Global control of linear systems with saturating actuators.
Automatica, 1998, 34(7): 897-905/
[12] D. A. Bini, B. Iannazzo, B. Meini, and F. Poloni. Nonsymmetric
algebraic Riccati equations associated with an M-matrix: recent
advances and algorithms. In Dagstuhl Seminar Proceedings, "Numerical
Methods for Structured Markov Chains", 07461, 2007.
[13] C.-H. Guo. Nonsymmetric algebraic Riccati equations and Wiener-Hopf
factorization for M-matrices. SIAM J. Matrix Anal. Appl., 23(1):225_242
(electronic), 2001.
[14] Ni M L. A note on the maximum solutions of Riccati equations.
Automatica, 1991, 27(6): 1059-1060.
[15] Ni Mao-Lin. Design of Robust Control Systems: Theory and
Applications (Ph. D. dissertation), Chinese Academy of Space
Technology, 1992 (in Chinese).
[16] Zhou K M, Doyle J, Glover K. Robust and Optimal Control. New
Jersey: Prentice-Hall, 1996.
[17] Kaveh, A. Structural Mechanics: Graph and Matrix Methods, 3rd ed.
Somerset: Research Studies Press, 2004.
[18] A. Kaveh and K. Koohestani, Combinatorial optimization of special
graphs for nodal ordering and graph partitioning, Acta Mechanica, Nos.
(1-2), 207(2009)95-108.
496-501.
[19] Potter, J. E. 1966. Matrix quadratic solutions. SIAM J. Appl. Math. 14:</p>
@article{"International Journal of Engineering, Mathematical and Physical Sciences:65313", author = "Mahdi Nouri", title = "Algebraic Riccati Matrix Equation for Eigen- Decomposition of Special Structured Matrices; Applications in Structural Mechanics", abstract = "In this paper Algebraic Riccati matrix equation is used for Eigen-decomposition of special structured matrices. This is achieved by similarity transformation and then using algebraic riccati matrix equation to triangulation of matrices. The process is decomposition of matrices into small and specially structured submatrices with low dimensions for fast and easy finding of Eigenpairs. Numerical and structural examples included showing the efficiency of present method.
", keywords = "Riccati, matrix equation, eigenvalue problem,
symmetric, bisymmetric, persymmetric, decomposition, canonical
forms, Graphs theory, adjacency and Laplacian matrices.", volume = "7", number = "3", pages = "501-5", }
